properties of special parallelograms - big ideas math · rectangles, rhombuses, and squares? ......

Section 7.4 Properties of Special Parallelograms 387

Properties of Special Parallelograms7.4

Essential QuestionEssential Question What are the properties of the diagonals of

rectangles, rhombuses, and squares?

Recall the three types of parallelograms shown below.

Rhombus Rectangle Square

Identifying Special Quadrilaterals

Work with a partner. Use dynamic geometry software.

a. Draw a circle with center A. Sample

b. Draw two diameters of the circle.

Label the endpoints B, C, D, and E.

c. Draw quadrilateral BDCE.

d. Is BDCE a parallelogram?

rectangle? rhombus? square?

Explain your reasoning.

e. Repeat parts (a)–(d) for several

other circles. Write a conjecture

based on your results.

Identifying Special Quadrilaterals

Work with a partner. Use dynamic geometry software.

a. Construct two segments that are Sampleperpendicular bisectors of each

other. Label the endpoints A, B, D,

and E. Label the intersection C.

b. Draw quadrilateral AEBD.

c. Is AEBD a parallelogram?

rectangle? rhombus? square?


d. Repeat parts (a)–(c) for several

other segments. Write a conjecture

based on your results.

Communicate Your AnswerCommunicate Your Answer 3. What are the properties of the diagonals of rectangles, rhombuses, and squares?

4. Is RSTU a parallelogram? rectangle? rhombus? square? Explain your reasoning.

5. What type of quadrilateral has congruent diagonals that bisect each other?

CONSTRUCTING VIABLE ARGUMENTS

To be profi cient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

A

C

D

E

B

A

C

D

E

B

S

TU

R

F

hs_geo_pe_0704.indd 387hs_geo_pe_0704.indd 387 1/19/15 11:53 AM1/19/15 11:53 AM

388 Chapter 7 Quadrilaterals and Other Polygons

7.4 Lesson What You Will LearnWhat You Will Learn Use properties of special parallelograms.

Use properties of diagonals of special parallelograms.

Use coordinate geometry to identify special types of parallelograms.

Using Properties of Special ParallelogramsIn this lesson, you will learn about three special types of parallelograms: rhombuses,

rectangles, and squares.

rhombus, p. 388rectangle, p. 388square, p. 388

Previousquadrilateralparallelogramdiagonal

Core VocabularyCore Vocabullarry

Core Core ConceptConceptRhombuses, Rectangles, and Squares

A rhombus is a

parallelogram with

four congruent sides.

A rectangle is a

parallelogram with

four right angles.

A square is a parallelogram

with four congruent sides

and four right angles.

You can use the corollaries below to prove that a quadrilateral is a rhombus, rectangle,

or square, without fi rst proving that the quadrilateral is a parallelogram.

CorollariesCorollariesCorollary 7.2 Rhombus CorollaryA quadrilateral is a rhombus if and only if it has

four congruent sides.

ABCD is a rhombus if and only if

— AB ≅ — BC ≅ — CD ≅ — AD .

Proof Ex. 81, p. 396

Corollary 7.3 Rectangle CorollaryA quadrilateral is a rectangle if and only if it has

four right angles.

ABCD is a rectangle if and only if

∠A, ∠B, ∠C, and ∠D are right angles.


Corollary 7.4 Square CorollaryA quadrilateral is a square if and only if it is

a rhombus and a rectangle.

ABCD is a square if and only if

— AB ≅ — BC ≅ — CD ≅ — AD and ∠A, ∠B, ∠C,

and ∠D are right angles.


A B

CD

A B

CD

A B

CD



The Venn diagram below illustrates some important relationships among

parallelograms, rhombuses, rectangles, and squares. For example, you can

see that a square is a rhombus because it is a parallelogram with four congruent

sides. Because it has four right angles, a square is also a rectangle.

parallelograms(opposite sides are parallel)

rhombuses(4 congruent sides)

rectangles(4 right angles)

squares

Using Properties of Special Quadrilaterals

For any rhombus QRST, decide whether the statement is always or sometimes true.

Draw a diagram and explain your reasoning.

a. ∠Q ≅ ∠S b. ∠Q ≅ ∠R

SOLUTION

a. By defi nition, a rhombus is a parallelogram with four

congruent sides. By the Parallelogram Opposite Angles

Theorem (Theorem 7.4), opposite angles of a parallelogram

are congruent. So, ∠Q ≅ ∠S. The statement is always true.

b. If rhombus QRST is a square, then all four angles are

congruent right angles. So, ∠Q ≅ ∠R when QRST is

a square. Because not all rhombuses are also squares,

the statement is sometimes true.

Classifying Special Quadrilaterals

Classify the special quadrilateral.


SOLUTION

The quadrilateral has four congruent sides. By the Rhombus Corollary, the

quadrilateral is a rhombus. Because one of the angles is not a right angle,

the rhombus cannot be a square.

Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com

1. For any square JKLM, is it always or sometimes true that — JK ⊥ — KL ? Explain

your reasoning.

2. For any rectangle EFGH, is it always or sometimes true that — FG ≅ — GH ?


3. A quadrilateral has four congruent sides and four congruent angles. Sketch the

quadrilateral and classify it.

Q R

ST

Q R

ST

70°



Part of Rhombus Diagonals Theorem

Given ABCD is a rhombus.

Prove — AC ⊥ — BD

ABCD is a rhombus. By the defi nition of a rhombus, — AB ≅ — BC . Because a rhombus is a parallelogram and the

diagonals of a parallelogram bisect each other, — BD bisects — AC at E. So, — AE ≅ — EC . — BE ≅ — BE by the Refl exive Property of Congruence (Theorem 2.1). So,

△AEB ≅ △CEB by the SSS Congruence Theorem (Theorem 5.8). ∠AEB ≅ ∠CEB because corresponding parts of congruent triangles are congruent. Then by the Linear

Pair Postulate (Postulate 2.8), ∠AEB and ∠CEB are supplementary. Two congruent

angles that form a linear pair are right angles, so m∠AEB = m∠CEB = 90° by the

defi nition of a right angle. So, — AC ⊥ — BD by the defi nition of perpendicular lines.

Finding Angle Measures in a Rhombus

Find the measures of the numbered angles in rhombus ABCD.

SOLUTION

Use the Rhombus Diagonals Theorem and the Rhombus Opposite Angles Theorem to

fi nd the angle measures.

m∠1 = 90° The diagonals of a rhombus are perpendicular.

m∠2 = 61° Alternate Interior Angles Theorem (Theorem 3.2)

m∠3 = 61° Each diagonal of a rhombus bisects a pair ofopposite angles, and m∠2 = 61°.

m∠1 + m∠3 + m∠4 = 180° Triangle Sum Theorem (Theorem 5.1)

90° + 61° + m∠4 = 180° Substitute 90° for m∠1 and 61° for m∠3.

m∠4 = 29° Solve for m∠4.

So, m∠1 = 90°, m∠2 = 61°, m∠3 = 61°, and m∠4 = 29°.

Using Properties of Diagonals

TheoremsTheoremsTheorem 7.11 Rhombus Diagonals TheoremA parallelogram is a rhombus if and only if its

diagonals are perpendicular.

▱ABCD is a rhombus if and only if — AC ⊥ — BD .

Proof p. 390; Ex. 72, p. 395

Theorem 7.12 Rhombus Opposite Angles TheoremA parallelogram is a rhombus if and only if each

diagonal bisects a pair of opposite angles.

▱ABCD is a rhombus if and only if — AC bisects ∠BCD

and ∠BAD, and — BD bisects ∠ABC and ∠ADC.

Proof Exs. 73 and 74, p. 395

READINGRecall that biconditionals, such as the Rhombus Diagonals Theorem, can be rewritten as two parts. To prove a biconditional, you must prove both parts.

C

BA

D

C

BA

D

C

BA

D61°

13 2

4

C

E

BA

D



Identifying a Rectangle

You are building a frame for a window. The window will be installed in the opening

shown in the diagram.

a. The opening must be a rectangle. Given the measurements in the diagram, can you

assume that it is? Explain.

b. You measure the diagonals of the opening. The diagonals are 54.8 inches and

55.3 inches. What can you conclude about the shape of the opening?

SOLUTION

a. No, you cannot. The boards on opposite sides are the same length, so they form a

parallelogram. But you do not know whether the angles are right angles.

b. By the Rectangle Diagonals Theorem, the diagonals of a rectangle are congruent.

The diagonals of the quadrilateral formed by the boards are not congruent, so the

boards do not form a rectangle.

Finding Diagonal Lengths in a Rectangle

In rectangle QRST, QS = 5x − 31 and RT = 2x + 11.

Find the lengths of the diagonals of QRST.

SOLUTION

By the Rectangle Diagonals Theorem, the diagonals of a rectangle are congruent. Find

x so that — QS ≅ — RT .

QS = RT Set the diagonal lengths equal.

5x − 31 = 2x + 11 Substitute 5x − 31 for QS and 2x + 11 for RT.

3x − 31 = 11 Subtract 2x from each side.

3x = 42 Add 31 to each side.

x = 14 Divide each side by 3.

When x = 14, QS = 5(14) − 31 = 39 and RT = 2(14) + 11 = 39.

Each diagonal has a length of 39 units.


4. In Example 3, what is m∠ADC and m∠BCD?

5. Find the measures of the numbered angles in

rhombus DEFG.

TheoremTheoremTheorem 7.13 Rectangle Diagonals TheoremA parallelogram is a rectangle if and only if

its diagonals are congruent.

▱ABCD is a rectangle if and only if — AC ≅ — BD .

Proof Exs. 87 and 88, p. 396

ED

G F118°

1

3

2

4

B

CD

A

33 in.

33 in.

44 in. 44 in.

33 in.

33 in.

44 in. 44 in.

R

ST

Q




6. Suppose you measure only the diagonals of the window opening in

Example 4 and they have the same measure. Can you conclude that the

opening is a rectangle? Explain.

7. WHAT IF? In Example 5, QS = 4x − 15 and RT = 3x + 8. Find the lengths of the

diagonals of QRST.

Using Coordinate Geometry

Identifying a Parallelogram in the Coordinate Plane

Decide whether ▱ABCD with vertices A(−2, 6), B(6, 8), C(4, 0), and D(−4, −2) is a

rectangle, a rhombus, or a square. Give all names that apply.

SOLUTION

1. Understand the Problem You know the vertices of ▱ABCD. You need to identify

the type of parallelogram.

2. Make a Plan Begin by graphing the vertices. From the graph, it appears that all

four sides are congruent and there are no right angles.

Check the lengths and slopes of the diagonals of ▱ABCD. If the diagonals are

congruent, then ▱ABCD is a rectangle. If the diagonals are perpendicular, then

▱ABCD is a rhombus. If they are both congruent and perpendicular, then ▱ABCD

is a rectangle, a rhombus, and a square.

3. Solve the Problem Use the Distance Formula to fi nd AC and BD.

AC = √——

(−2 − 4)2 + (6 − 0)2 = √—

72 = 6 √—

2

BD = √———

[6 − (−4)]2 + [8 − (−2)]2 = √—

200 =10 √—

2

Because 6 √—

2 ≠ 10 √—

2 , the diagonals are not congruent. So, ▱ABCD is not a

rectangle. Because it is not a rectangle, it also cannot be a square.

Use the slope formula to fi nd the slopes of the diagonals — AC and — BD .

slope of — AC = 6 − 0

— −2 − 4

= 6 —

−6 = −1 slope of — BD =

8 − (−2) —

6 − (−4) =

10 —

10 = 1

Because the product of the slopes of the diagonals is −1, the diagonals are

perpendicular.

So, ▱ABCD is a rhombus.

4. Look Back Check the side lengths of ▱ABCD. Each side has a length of 2 √—

17

units, so ▱ABCD is a rhombus. Check the slopes of two consecutive sides.

slope of — AB = 8 − 6

— 6 − (−2)

= 2 —

8 =

1 —

4 slope of — BC =

8 − 0 —

6 − 4 =

8 —

2 = 4

Because the product of these slopes is not −1, — AB is not perpendicular to — BC .

So, ∠ABC is not a right angle, and ▱ABCD cannot be a rectangle or a square. ✓


8. Decide whether ▱PQRS with vertices P(−5, 2), Q(0, 4), R(2, −1), and

S(−3, −3) is a rectangle, a rhombus, or a square. Give all names that apply.

x

y

8

4

−4

−8

A(−2, 6)

D(−4, −2)

B(6, 8)

C(4, 0)



Exercises7.4 Dynamic Solutions available at BigIdeasMath.com

In Exercises 3–8, for any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning. (See Example 1.)

3. ∠L ≅ ∠M 4. ∠K ≅ ∠M

5. — JM ≅ — KL 6. — JK ≅ — KL

7. — JL ≅ — KM 8. ∠JKM ≅ ∠LKM

In Exercises 9–12, classify the quadrilateral. Explain your reasoning. (See Example 2.)

9. 10.

11. 12.

In Exercises 13–16, fi nd the measures of the numbered angles in rhombus DEFG. (See Example 3.)

13.

E

DG

F27°

1 32

456

14. ED

G F

48°

1

32

4

5

15. E

D

GF

106°

1

3

2

4

5

16.

In Exercises 17–22, for any rectangle WXYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning.

17. ∠W ≅ ∠X 18. — WX ≅ — YZ

19. — WX ≅ — XY 20. — WY ≅ — XZ

21. — WY ⊥ — XZ 22. ∠WXZ ≅ ∠YXZ

In Exercises 23 and 24, determine whether the quadrilateral is a rectangle. (See Example 4.)

23.

24.

21 m21 m

32 m

32 m

In Exercises 25–28, fi nd the lengths of the diagonals of rectangle WXYZ. (See Example 5.)

25. WY = 6x − 7 26. WY = 14x + 10

XZ = 3x + 2 XZ = 11x + 22

27. WY = 24x − 8 28. WY = 16x + 2

XZ = −18x + 13 XZ = 36x − 6

Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics

1. VOCABULARY What is another name for an equilateral rectangle?

2. WRITING What should you look for in a parallelogram to know if the parallelogram is also a rhombus?

Vocabulary and Core Concept CheckVocabulary and Core Concept Check

10

. 12.

140°

140°

40°

E

D

G

F

72°1

32

4

5



In Exercises 29–34, name each quadrilateral —parallelogram, rectangle, rhombus, or square—for which the statement is always true.

29. It is equiangular.

30. It is equiangular and equilateral.

31. The diagonals are perpendicular.

32. Opposite sides are congruent.

33. The diagonals bisect each other.

34. The diagonals bisect opposite angles.

35. ERROR ANALYSIS Quadrilateral PQRS is a rectangle.

Describe and correct the error in fi nding the value of x.

58°

x°

P Q

RS

m∠QSR = m∠QSP x ° = 58° x = 58

✗

36. ERROR ANALYSIS Quadrilateral PQRS is a rhombus.

Describe and correct the error in fi nding the value of x.

37°x°

P Q

S R

m∠QRP = m∠SQR x° = 37° x = 37

✗

In Exercises 37–42, the diagonals of rhombus ABCD intersect at E. Given that m∠BAC = 53°, DE = 8, and EC = 6, fi nd the indicated measure.

53°

8 6

BA

C

E

D

37. m∠DAC 38. m∠AED

39. m∠ADC 40. DB

41. AE 42. AC

In Exercises 43–48, the diagonals of rectangle QRST intersect at P. Given that m∠PTS = 34° and QS = 10, fi nd the indicated measure.

34°

R

P

ST

Q

43. m∠QTR 44. m∠QRT

45. m∠SRT 46. QP

47. RT 48. RP

In Exercises 49–54, the diagonals of square LMNP intersect at K. Given that LK = 1, fi nd the indicated measure.

1

M

K

NP

L

49. m∠MKN 50. m∠LMK

51. m∠LPK 52. KN

53. LN 54. MP

In Exercises 55–60, decide whether ▱JKLM is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning. (See Example 6.)

55. J(−4, 2), K(0, 3), L(1, −1), M(−3, −2)

56. J(−2, 7), K(7, 2), L(−2, −3), M(−11, 2)

57. J(3, 1), K(3, −3), L(−2, −3), M(−2, 1)

58. J(−1, 4), K(−3, 2), L(2, −3), M(4, −1)

59. J(5, 2), K(1, 9), L(−3, 2), M(1, −5)

60. J(5, 2), K(2, 5), L(−1, 2), M(2, −1)

MATHEMATICAL CONNECTIONS In Exercises 61 and 62, classify the quadrilateral. Explain your reasoning. Then fi nd the values of x and y.

61. y + 8 3y

104°

x°A C

B

D

62.

(3x + 18)°2y

10

5x°Q R

SP



63. DRAWING CONCLUSIONS In the window,— BD ≅ — DF ≅ — BH ≅ — HF . Also, ∠HAB, ∠BCD,

∠DEF, and ∠FGH are right angles.

A B C

D

A B C

G F E

HJ

G F E

HJ

D

a. Classify HBDF and ACEG. Explain your

reasoning.

b. What can you conclude about the lengths of the

diagonals — AE and — GC ? Given that these diagonals

intersect at J, what can you conclude about the

lengths of — AJ , — JE , — CJ , and — JG ? Explain.

64. ABSTRACT REASONING Order the terms in a diagram

so that each term builds off the previous term(s).

Explain why each fi gure is in the location you chose.

quadrilateral

rhombus

parallelogram

rectangle

square

CRITICAL THINKING In Exercises 65–70, complete each statement with always, sometimes, or never. Explain your reasoning.

65. A square is _________ a rhombus.

66. A rectangle is _________ a square.

67. A rectangle _________ has congruent diagonals.

68. The diagonals of a square _________ bisect

its angles.

69. A rhombus _________ has four congruent angles.

70. A rectangle _________ has perpendicular diagonals.

71. USING TOOLS You want to mark off a square region

for a garden at school. You use a tape measure to

mark off a quadrilateral on the ground. Each side

of the quadrilateral is 2.5 meters long. Explain how

you can use the tape measure to make sure that the

quadrilateral is a square.

72. PROVING A THEOREM Use the plan for proof

below to write a paragraph proof for one part of the

Rhombus Diagonals Theorem (Theorem 7.11).

C

X

BA

D

Given ABCD is a parallelogram.— AC ⊥ — BD

Prove ABCD is a rhombus.

Plan for Proof Because ABCD is a parallelogram,

its diagonals bisect each other at X. Use — AC ⊥ — BD to show that △BXC ≅ △DXC. Then show that — BC ≅ — DC. Use the properties of a parallelogram to

show that ABCD is a rhombus.

PROVING A THEOREM In Exercises 73 and 74, write a proof for part of the Rhombus Opposite Angles Theorem (Theorem 7.12).

73. Given PQRS is a parallelogram.— PR bisects ∠SPQ and ∠QRS.— SQ bisects ∠PSR and ∠RQP.

Prove PQRS is a rhombus.

S

TR

Q

P

74. Given WXYZ is a rhombus.

Prove — WY bisects ∠ZWX and ∠XYZ.— ZX bisects ∠WZY and ∠YXW.

W X

YZ

V



75. ABSTRACT REASONING Will a diagonal of a square

ever divide the square into two equilateral triangles?


76. ABSTRACT REASONING Will a diagonal of a rhombus

ever divide the rhombus into two equilateral triangles?


77. CRITICAL THINKING Which quadrilateral could be

called a regular quadrilateral? Explain your reasoning.

78. HOW DO YOU SEE IT? What other information do

you need to determine whether the fi gure is

a rectangle?

79. REASONING Are all rhombuses similar? Are all

squares similar? Explain your reasoning.

80. THOUGHT PROVOKING Use the Rhombus Diagonals

Theorem (Theorem 7.11) to explain why every

rhombus has at least two lines of symmetry.

PROVING A COROLLARY In Exercises 81–83, write the corollary as a conditional statement and its converse. Then explain why each statement is true.

81. Rhombus Corollary (Corollary 7.2)

82. Rectangle Corollary (Corollary 7.3)

83. Square Corollary (Corollary 7.4)

84. MAKING AN ARGUMENT Your friend claims a

rhombus will never have congruent diagonals because

it would have to be a rectangle. Is your friend correct?


85. PROOF Write a proof in the style of your choice.

Given △XYZ ≅ △XWZ, ∠XYW ≅ ∠ZWY

Prove WXYZ is a rhombus.

X Y

ZW

86. PROOF Write a proof in the style of your choice.

Given — BC ≅ — AD , — BC ⊥ — DC , — AD ⊥ — DC

Prove ABCD is a rectangle.

B

CD

A

PROVING A THEOREM In Exercises 87 and 88, write a proof for part of the Rectangle Diagonals Theorem (Theorem 7.13).

87. Given PQRS is a rectangle.

Prove — PR ≅ — SQ

88. Given PQRS is a parallelogram.— PR ≅ — SQ

Prove PQRS is a rectangle.

Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency — DE is a midsegment of △ABC. Find the values of x and y. (Section 6.4)

89.

B

C

D

E

A

10

16

1212

x y

90.

B C

D E

A

6

7

x

y

91.

BC

D

E

A

99

13

x

y

Reviewing what you learned in previous grades and lessons


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